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Diagonalization is the process of finding a diagonal matrix that is similar to a given square matrix.

A diagonal matrix is a matrix where the elements outside the main diagonal are all zero.

A matrix is diagonalizable if there are enough linearly independent eigenvectors to form a basis for its column space.

An eigenvalue is a scalar \( \lambda \) such that \( A \mathbf{v} = \lambda \mathbf{v} \) for some nonzero vector \( \mathbf{v} \) and square matrix A.

An eigenvector is a nonzero vector \( \mathbf{v} \) that, when multiplied by a matrix, results in a scalar multiple of itself, \( A \mathbf{v} = \lambda \mathbf{v} \).

The characteristic polynomial of a matrix A is the polynomial given by the determinant \( \det(A - \lambda I) \), where I is the identity matrix and \( \lambda \) is a scalar.

Two matrices A and B are similar if there exists an invertible matrix P such that \( P^{-1}AP = B \).

A matrix must have n linearly independent eigenvectors to be diagonalizable, where n is the size of the matrix.

The diagonal matrix contains the eigenvalues of the original matrix along its main diagonal, in the same order as the corresponding eigenvectors in the columns of the matrix used to diagonalize.

The eigenspace of an eigenvalue is the set of all eigenvectors corresponding to that eigenvalue, along with the zero vector.

The geometric multiplicity of an eigenvalue is the dimension of its eigenspace, or the number of linearly independent eigenvectors associated with that eigenvalue.

The spectral theorem states that a matrix is diagonalizable using an orthonormal basis of eigenvectors if and only if the matrix is normal (commutes with its conjugate transpose).

The algebraic multiplicity of an eigenvalue is the number of times the eigenvalue appears as a root of the characteristic polynomial.

The determinant can help in finding eigenvalues as it is used in the characteristic polynomial, which is solved to get eigenvalues.

The trace of a matrix is the sum of its diagonal entries and is equal to the sum of its eigenvalues considering their algebraic multiplicities.

Matrix diagonalization simplifies the computation of matrix powers, as powering a diagonal matrix involves simply raising each diagonal element to the power.

If a matrix has complex eigenvalues, it is not diagonalizable over the real numbers, but it can be diagonalized over the complex numbers.

The exponential of a diagonalizable matrix simplifies to the exponentials of the individual eigenvalues along the main diagonal of the diagonal matrix.

A defective matrix is a matrix that does not have enough linearly independent eigenvectors to be diagonalizable.